Let $R$ be a ring with unity, $M$ be a right $R$-module and $\text{End}_R(M)$ be the endomorphism ring of $M$. Suppose that $\text{End}_R(M)$ is a reduced ring, that is, $f^2=0$ implies $f=0$ for every $\text{End}_R(M)$.
Question: Is it right to conclude that for any $0\neq m\in M,$ $f^2(m)=0$ implies $f(m)=0$?
My thinking is that it is right since $f^2(m)=0$ for $m\neq 0$ implies $f^2=0$. Since $f=0$ by hypothesis, $f(m)=0$.